Subgroups — C₃₁×#14714 #21471473

Order 7936.

2078 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#391	C₂	1	4	yes	maximal
3	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×C₄⋊#95	C₂	1	8	yes	maximal
6	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#391	—	2	16	no	—
17	C₃₁×#384	—	2	4	no	—
18	C₂×C₃₁×#95	C₂²	1	16	yes	—
19	C₄³×C₃₁	—	2	4	no	—
20	C₂²×Q₈×C₃₁	—	2	4	no	—
21	C₂²×Q₈×C₃₁	—	2	4	no	—
22	C₂²×Q₈×C₃₁	—	2	8	no	—
23	C₂²×Q₈×C₃₁	—	2	8	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
30	C₃₁×C₄⋊Q₈	—	2	32	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
38	C₂×C₄²×C₃₁	—	2	4	no	—
39	C₂×C₄²×C₃₁	—	2	8	no	—
40	C₂×C₄²×C₃₁	—	2	4	no	—
41	C₂×C₄²×C₃₁	—	2	8	no	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₃₁	D₄	1	8	yes	—
44	C₂×C₄²×C₃₁	D₄	1	4	yes	—
45	C₂×C₄²×C₃₁	D₄	1	4	yes	—
46	C₂×C₄²×C₃₁	Q₈	1	16	yes	complemented
47	C₃₁×#95	—	2	32	no	—
48	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	—	2	16	no	—
50	C₂×Q₈×C₃₁	—	2	32	no	—
51	C₂×Q₈×C₃₁	—	4	16	no	—
52	C₂×Q₈×C₃₁	—	4	8	no	—
53	C₂×Q₈×C₃₁	—	4	8	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	8	no	—
57	C₂²×C₄×C₃₁	—	2	8	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	8	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	2	16	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₃₁	C₂×Q₈	1	16	yes	—
74	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
75	C₂²×C₄×C₃₁	#41	1	4	yes	—
76	C₂²×C₄×C₃₁	#41	1	8	yes	—
77	C₄⋊C₄×C₃₁	—	2	32	no	—
78	C₄⋊C₄×C₃₁	—	4	16	no	—
79	C₄⋊C₄×C₃₁	—	4	8	no	—
80	C₄²×C₃₁	—	2	16	no	—
81	C₄²×C₃₁	—	2	32	no	—
82	C₄²×C₃₁	C₂²⋊C₄	1	16	yes	—
83	#14714	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
88	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
89	C₂³×C₃₁	C₂×#41	1	2	yes	—
90	C₂³×C₃₁	C₂×#41	1	4	yes	—
91	Q₈×C₃₁	—	4	32	no	—
92	C₂×C₄×C₃₁	—	2	8	no	—
93	C₂×C₄×C₃₁	—	2	16	no	—
94	C₂×C₄×C₃₁	—	2	8	no	—
95	C₂×C₄×C₃₁	—	2	16	no	—
96	C₂×C₄×C₃₁	—	2	8	no	—
97	C₂×C₄×C₃₁	—	2	16	no	—
98	C₂×C₄×C₃₁	—	4	8	no	—
99	C₂×C₄×C₃₁	—	4	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	8	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	16	no	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
106	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₃₁	C₄×Q₈	1	16	yes	—
108	C₂×C₄×C₃₁	#121	1	8	yes	—
109	C₂×C₄×C₃₁	C₂²⋊Q₈	1	16	yes	—
110	C₂×C₄×C₃₁	#124	1	4	yes	—
111	C₂×C₄×C₃₁	#124	1	8	yes	—
112	C₂×C₄×C₃₁	#127	1	4	yes	—
113	C₂×C₄×C₃₁	C₄⋊Q₈	1	8	yes	—
114	C₂×C₄×C₃₁	C₄⋊Q₈	1	16	yes	complemented
115	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
116	C₂×#391	C₂×C₃₁	1	4	yes	—
117	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
118	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
119	C₄⋊#95	C₂×C₃₁	1	8	yes	—
120	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
122	C₂²×C₃₁	C₂×C₄×Q₈	1	4	yes	—
123	C₂²×C₃₁	C₂×#121	1	2	yes	—
124	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
125	C₂²×C₃₁	C₂×#124	1	2	yes	—
126	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
127	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×C₄⋊Q₈	1	4	yes	—
129	C₂²×C₃₁	C₂×C₄⋊Q₈	1	2	yes	—
130	C₂²×C₃₁	#532	1	4	yes	—
131	C₂²×C₃₁	#533	1	8	yes	—
132	C₄×C₃₁	—	2	16	no	—
133	C₄×C₃₁	—	4	8	no	—
134	C₄×C₃₁	—	8	4	no	—
135	C₄×C₃₁	#390	1	8	yes	—
136	C₄×C₃₁	#391	1	16	yes	complemented
137	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
138	C₂×C₄⋊Q₈	—	2	4	no	—
139	C₂×C₄⋊Q₈	—	2	4	no	—
140	C₂×C₄⋊Q₈	C₄×C₃₁	1	8	yes	complemented
141	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
144	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
145	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
146	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
147	#391	—	2	16	no	—
148	#384	—	2	4	no	—
149	C₂×#95	C₂²×C₃₁	1	16	yes	—
150	C₄³	—	2	4	no	—
151	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#391	1	2	yes	—
153	C₂×C₃₁	C₄×C₄⋊Q₈	1	4	yes	—
154	C₂×C₃₁	#2857	1	4	yes	—
155	C₂×C₃₁	#2858	1	1	yes	characteristic
156	C₂×C₃₁	#2862	1	2	yes	—
157	C₂×C₃₁	#2868	1	1	yes	characteristic
158	C₂²×Q₈	—	2	4	no	—
159	C₂²×Q₈	—	2	4	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
163	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
164	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
165	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
166	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
168	C₄⋊Q₈	—	2	32	no	—
169	C₂×C₄⋊C₄	—	2	8	no	—
170	C₂×C₄⋊C₄	—	2	8	no	—
171	C₂×C₄⋊C₄	—	2	8	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	—	2	16	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	2	8	no	—
178	C₂×C₄²	—	2	4	no	—
179	C₂×C₄²	—	2	8	no	—
180	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
181	C₂×C₄²	D₄×C₃₁	1	8	yes	—
182	C₂×C₄²	D₄×C₃₁	1	4	yes	—
183	C₂×C₄²	D₄×C₃₁	1	4	yes	—
184	C₂×C₄²	Q₈×C₃₁	1	16	yes	complemented
185	#95	—	2	32	no	—
186	C₃₁	#14714	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
188	C₂×Q₈	—	2	16	no	—
189	C₂×Q₈	—	2	32	no	—
190	C₂×Q₈	—	4	16	no	—
191	C₂×Q₈	—	4	8	no	—
192	C₂×Q₈	—	4	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
204	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
206	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
207	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
211	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
212	C₂²×C₄	C₂×Q₈×C₃₁	1	16	yes	—
213	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
214	C₂²×C₄	#41×C₃₁	1	4	yes	—
215	C₂²×C₄	#41×C₃₁	1	8	yes	—
216	C₄⋊C₄	—	2	32	no	—
217	C₄⋊C₄	—	4	16	no	—
218	C₄⋊C₄	—	4	8	no	—
219	C₄²	—	2	16	no	—
220	C₄²	—	2	32	no	—
221	C₄²	C₂²⋊C₄×C₃₁	1	16	yes	—
222	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
223	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
224	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
225	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
226	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
227	C₂³	C₂×#41×C₃₁	1	2	yes	—
228	C₂³	C₂×#41×C₃₁	1	4	yes	—
229	Q₈	—	4	32	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
244	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
245	C₂×C₄	C₄×Q₈×C₃₁	1	16	yes	—
246	C₂×C₄	C₃₁×#121	1	8	yes	—
247	C₂×C₄	C₃₁×C₂²⋊Q₈	1	16	yes	—
248	C₂×C₄	C₃₁×#124	1	4	yes	—
249	C₂×C₄	C₃₁×#124	1	8	yes	—
250	C₂×C₄	C₃₁×#127	1	4	yes	—
251	C₂×C₄	C₃₁×C₄⋊Q₈	1	8	yes	—
252	C₂×C₄	C₃₁×C₄⋊Q₈	1	16	yes	complemented
253	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
255	C₂²	C₂×C₄×Q₈×C₃₁	1	4	yes	—
256	C₂²	C₂×C₃₁×#121	1	2	yes	—
257	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
258	C₂²	C₂×C₃₁×#124	1	2	yes	—
259	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
260	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
261	C₂²	C₂×C₃₁×C₄⋊Q₈	1	4	yes	—
262	C₂²	C₂×C₃₁×C₄⋊Q₈	1	2	yes	—
263	C₂²	C₃₁×#532	1	4	yes	—
264	C₂²	C₃₁×#533	1	8	yes	—
265	C₄	—	2	16	no	—
266	C₄	—	4	8	no	—
267	C₄	—	8	4	no	—
268	C₄	C₃₁×#390	1	8	yes	—
269	C₄	C₃₁×#391	1	16	yes	complemented
270	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
271	C₂	C₂×C₃₁×#391	1	2	yes	—
272	C₂	C₄×C₃₁×C₄⋊Q₈	1	4	yes	—
273	C₂	C₃₁×#2857	1	4	yes	—
274	C₂	C₃₁×#2858	1	1	yes	characteristic
275	C₂	C₃₁×#2862	1	2	yes	—
276	C₂	C₃₁×#2868	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
46. C₂×C₄²×C₃₁	Q₈	semidirect
83. #14714	C₃₁	direct (N × H)
114. C₂×C₄×C₃₁	C₄⋊Q₈	semidirect
136. C₄×C₃₁	#391	semidirect
140. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
184. C₂×C₄²	Q₈×C₃₁	semidirect
186. C₃₁	#14714	direct (N × H)
252. C₂×C₄	C₃₁×C₄⋊Q₈	semidirect
269. C₄	C₃₁×#391	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₂×C₃₁×#391	2
C₂×C₃₁×#384	2
C₂×C₄³×C₃₁	2
C₃₁×C₄⋊#95	2
#14714	31

Subgroups — C31×#14714 #21471473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14714 #21471473